On the Syntactic Monoids Associated with a Class of Synchronized Codes

A complete code C over an alphabet A is called synchronized if there exist x, y ∈ C* such that xA*∩A*y⊆C*. In this paper we describe the syntactic monoid Syn(C +) of C + for a complete synchronized code C over A such that C +, the semigroup generated by C, is a single class of its syntactic congruence P C+. In particular, we prove that, for such a code C, either C = A or Syn(C +) is isomorphic to a special submonoid of 𝒯 l(I) × 𝒯 r(Λ), where 𝒯 l(I) and 𝒯 r(Λ) are the full transformation semigroups on the nonempty sets I and Λ, respectively.


Introduction
Theory of codes is an important branch in the field of information science. Many methods, including combinatorics methods, analysis methods, and algebraic methods, are applied to study codes. As a kind of algebraic methods, it is effective to study some kinds of codes by considering syntactic monoids of the semigroups and monoids generated by these codes.
As we have known, prefix codes have fundamental importance in theory of codes. Many authors are devoted to the investigation of prefix codes by using several methods (cf. [1][2][3][4]). In particular, Petrich et al. [4] investigated maximal prefix codes by considering the syntactic monoids of the semigroups generated by them in 1996. They described the semigroup structure of the syntactic monoid Syn( + ) of + , the semigroup generated by a maximal prefix code for which + is a single class of the syntactic congruence + .
On the other hand, synchronized codes are also important both in theory and in applications. Many interesting results are obtained on this class of codes in the text of Berstel et al. [1]. Recently, Liu [3] investigated synchronized codes by algebraic methods and obtained an algebraic characterization of complete synchronized codes (see Lemma 5 in this paper). Furthermore, Liu [3,5] also studied some generalizations of synchronized codes. In this paper, by using the algebraic characterization of complete synchronized codes obtained in Liu [3] and some techniques developed in Petrich et al. [4], we give a description of the syntactic monoid Syn( + ) of + for a complete synchronized code over an alphabet such that + is a single class of its syntactic congruence + .

Preliminaries
A semigroup is a left zero semigroup if = for any , ∈ . Dually, we have right zero semigroups. A rectangular band is a semigroup which is isomorphic to a direct product of a left zero semigroup and a right zero semigroup. For rectangular bands, we have the following obvious result.
An ideal of a semigroup is a nonempty subset of satisfying that the union of and is contained in . Recall that the unique minimum ideal (with respect to set inclusion) of a semigroup (if exists) is called the kernel of . For the kernel of a semigroup, we have the following.
Lemma 2 (see [6]). If the kernel of a semigroup only consists of idempotents, then this kernel is a rectangular band. Let be an element of a semigroup . Then the function defined by = for all ∈ is the inner left translation induced by . Dually, we have inner right translation induced by . Finally, the pair = ( , ) is the inner bitranslation induced by . The set Π( ) of all inner bitranslations is the inner part of Ω( ). From Corollary III.1.7 in Petrich [7], Π( ) is an ideal of Ω( ).
In the sequel, the set of all transformations on a set written and composed as right (resp., left) operators is denoted by T ( ) (resp., T ( )). The identity mapping on a set is denoted by . If ∈ , then ⟨ ⟩ denotes the constant function on whose value is . Clearly, T ( ) and T ( ) are semigroups with their own compositions and T 0 ( ) = {⟨ ⟩ | ∈ } (as left operators) and T 0 ( ) = {⟨ ⟩ | ∈ } (as right operators) are subsemigroups of T ( ) and T ( ), respectively.
On the translation hull of a rectangular band, we have the following results which can be found in Section III.7 in Petrich and Reilly [6].
Let be a semigroup and an ideal of . Then is called an extension of . From Definition III.5.4 in Petrich [7], an extension of is called dense if for each congruence on , the fact that the restriction of to is the equality relation on implies that is the equality relation on . On dense extensions of rectangular bands, the following results can be obtained as a special case from Theorem III.1.12 and Corollary III.5.5 in [7] and can also be proved easily.

Lemma 4. If is a dense extension of a rectangular band , then the following semigroup homomorphism
is injective, where for each ∈ , Clearly in this case, is isomorphic to which contains Π( ) as an ideal.
Let be a semigroup and let 1 be the semigroup obtained from by adjoining an identity if necessary. The syntactic congruence determined by a subset of is the following relation on : {( , ) ∈ × | V ∈ if and only if V ∈ for all , V ∈ 1 }. In particular, if = { }, we call { } the syntactic congruence determined by and denote it by . Moreover, is called disjunctive in if is the equality relation on . It is easy to see that the relation saturates for every subset of ; that is, is a union of some -classes for every subset of .
Let be an alphabet, let * be the free monoid generated by , and let 1 be the identity of * . For any ⊆ * , the quotient monoid * / is called the syntactic monoid of , to be denoted by Syn( ). A nonempty set of + = * \ {1} is called a code over if the fact that implies that = and = , = 1, 2, . . . , .
A submonoid of a monoid is called stable in if the fact that , , ∈ implies that ∈ for all , V, ∈ . It is well known that the monoid * generated by a code over is stable in * (see Proposition 2.

in [1]).
A code over is called complete if, for any ∈ * , there exist , V ∈ * such that V ∈ * , where * is the monoid generated by . Recall that a complete code over is said to be synchronized if * ∩ * ⊆ * for some , ∈ * (see details in Proposition 10.1.14 of [1]). On complete synchronized codes, Liu [3] obtained the following algebraic characterizations recently.
Lemma 5 (see [3]). A complete code over is synchronized if and only if the kernel of Syn( * ) is a rectangular band.

A Characterization of Complete Synchronized Codes
This section gives a characterization of complete synchronized codes by using the syntactic monoid Syn( + ) of + , the semigroup generated by a code over . To this aim, we need several lemmas.  Proof. (1) This follows from the fact that V ∈ + if and only if V ∈ * for any ∈ + and , V ∈ * .
(2) Let ∈ * and + 1. Since 1 ∉ + and + is a union of some + -classes, it follows that ∉ + . On the other hand, for any ∈ , we have + and + , whence , , ∈ + by the fact that + is a union of some +classes. Since is a code over , * is stable. Therefore, ∈ * . This implies that = 1. Thus, the + -class containing 1 is {1}.
If the * -class containing 1 is not {1}, then there exists ∈ + such that 1 * . Moreover, Syn( + ) \ {1 + } is a subsemigroup of Syn( + ) by item (3) of Lemma 6. In the sequel, we show that the following is a semigroup isomorphism. We first show that is well defined. In fact, let , ∈ * and * = * . We divide the discussion into the following four cases. (iv) ̸ = 1, ̸ = 1. This follows from item (1) in Lemma 6.
By similar methods, we can show that is injective, and the surjectivity of is obvious.
On the other hand, for any * , * ∈ Syn( * ), we assert that and so is a semigroup morphism. In fact, we have the following cases.

Lemma 8. If is a monoid with identity 1 and \ {1} is a subsemigroup of , then has a kernel if and only if \ {1}
has a kernel. If this is the case, the two kernels are equal.
the result follows.

Proof. Let
= Syn( + ) and denote the set of + -classes with representatives from by for ⊆ * . Since + is a + -class, we can also let = + . Obviously, is an idempotent in .
Now, let = . We assert that is the kernel of . In fact, is an ideal of clearly. Moreover, since is complete, there exist , V ∈ * such that V ∈ + for all ∈ * . Therefore, there exist , ∈ such that = for all ∈ . Now, let be an ideal of and ∈ . Then V = for some , V ∈ whence = = V ⊆ . Thus, is the least ideal of and so is the kernel of . By Theorem 9, is a rectangular band. If = { }, then we have = = for any ∈ . This implies that = = for all ∈ and ∈ * . Since + is a single + -class, it follows that , , ∈ + . Because * is stable, we have ∈ * . Therefore, * = * and hence = . A contradiction. Thus, ̸ = { }. Now let be a congruence on whose restriction to is the identity relation on . Assume that ∈ and . Then , where , , ∈ and thus = = . Since { , 1} is stable in , it follows that ∈ { , 1}. If = 1, then 1 and for any ∈ , we obtain , , and , , ∈ , whence = = . Thus, is a rectangular band with the identity . And hence, = { } by Lemma 1. A contradiction. It follows that = and { } is a -class in . Now, let , ∈ and . Then for any , V ∈ , we have V V. Observe that { } is a -class, it follows that V = if and only if V = . Therefore, in . By the disjunctiveness of in , we have = . We conclude that is the equality relation on . Thus, is a dense extension of .
We end our paper by giving an example to illustrate our result.